Abstract
We show that the 1-planar slope number of 3-connected cubic 1-planar graphs is at most 4 when edges are drawn as polygonal curves with at most 1 bend each. This bound is obtained by drawings whose vertex and crossing resolution is at least \(\pi /4\). On the other hand, if the embedding is fixed, then there is a 3-connected cubic 1-planar graph that needs 3 slopes when drawn with at most 1 bend per edge. We also show that 2 slopes always suffice for 1-planar drawings of subcubic 1-planar graphs with at most 2 bends per edge. This bound is obtained with vertex resolution \(\pi /2\) and the drawing is RAC (crossing resolution \(\pi /2\)). Finally, we prove lower bounds for the slope number of straight-line 1-planar drawings in terms of number of vertices and maximum degree.
Highlights
A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once
Despite the efforts made in the study of 1-planar graphs, only few results are known concerning their geometric representations
We study the existence of 1-planar drawings that simultaneously satisfy the following properties: edges are polylines using few bends and few distinct slopes for their segments, edge crossings occur at large angles, and pairs of edges incident to the same vertex form large angles
Summary
A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. We prove upper and lower bounds on the k-bend 1-planar slope number of 1-planar graphs, when k ∈ {0, 1, 2}. These bounds on the number of slopes and on the vertex/crossing resolution are clearly worst-case optimal.
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